Computation of specular reflections on a sphere: Assessment and validation of algorithms based on special boundary cases

Abstract

The calculation of specular reflections on a sphere is a common problem in the Geosciences in general and in Earth remote sensing in particular. Especially in Global Navigation Satellite System Reflectometry (GNSS-R), the interferometric or reflection-minus-direct radio propagation delay should be calculated to retrieve sea level in GNSS-R altimetry. However, the solution is not trivial, as can be attested by multiple algorithms reported in the literature, many reinvented independently, without the emergence of a preferred or standard algorithm. We propose their assessment and validation based on special boundary cases, at the spherical horizon and at the vertical direction (zenith), where independent closed-form trigonometric expressions exist. We implemented five different algorithms to compute spherical specular reflection points, including three analytical ones based on a quartic polynomial and two iterative methods. We also computed derived parameters, such as interferometric delay, grazing angle, slant distance, and arc length. We performed simulations for all algorithms considering one hundred different antenna heights above the surface, between 10 m and 1 km. Finally, we compared analytical and iterative algorithms results with truth values at the special boundary cases. On the horizon, we found that all algorithms presented negligible root-mean-square error (RMSE) only for interferometric delay, but all other parameters exhibited varying accuracies for each algorithm. The iterative methods had the worst results at the spherical horizon, reaching meter-level RMSE for reflection point position and slant distance, whereas Miller & Vegh algorithm had worst results at zenith, especially in terms of grazing angle, position coordinate and arc length. We found the algorithm of Fujimura et al. to be the most stable and accurate. We performed a relative intercomparison between Fujimura et al. and each other algorithms at 45° as an intermediary case. We confirmed Fermat's iterative algorithm had the worst performance. The algorithms of Martín-Neira and Helm exhibited a systematic effect on certain parameters, proportional to antenna height and inversely proportional to satellite elevation. Therefore, Fujimura et al. is the recommended algorithm to compute specular reflections on a sphere.